Harold Calvin Marston Morse

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Marston Morse, 1965

Harold Calvin Marston Morse , or Marston Morse for short , (born March 24, 1892 in Waterville , Maine , † June 22, 1977 in Princeton , New Jersey ) was an American mathematician who worked in the field of the calculus of variations and differential geometry .

Life

Marston Morse studied at Colby College in Waterville and then went to Harvard University , where he received his diploma in 1915 and his doctorate in 1917 with George David Birkhoff with a thesis on geodesics on surfaces of negative curvature. During the First World War he served with distinction in an ambulance unit in France. He then taught from 1919 at Harvard, at Cornell University (1920-1925), Brown University 1925/26, and then again at Harvard, before he went to the Institute for Advanced Study at Princeton in 1935 . During the Second World War, he wrote about 80 mathematical reports at the Institute for Advanced Study on questions of ballistics , the effect of ammunition and radar . In 1962 he retired.

His field of work was predominantly the calculus of variations "on a large scale" (global analysis). Here he created his own theory of the maxima and minima of functions, Morse theory , which has applications in numerous areas from mathematical physics, the theory of differential equations to differential topology.

Preparatory work on the Morse Code theory was done in the 19th century by James Clerk Maxwell ("On hills and dales", 1870) and Arthur Cayley ( On contour and slope line 1859) - the so-called "mountaineering formula": number of peaks plus number of valleys minus Number of passes equals two (the elementary proof can be provided by counting with rising or falling sea level). A height function f is assigned to a manifold ( Morse function ) and the critical points in which the gradient (derivative) of f vanishes are considered. These can be maxima, minima or saddle points. A Morse index is assigned to each critical point , which corresponds to the number of independent directions in which the function f decreases (i.e. for areas im at maxima 2, minima 0, saddle points 1). The whole thing can be formalized: in critical points the first derivative disappears. If the matrix of the 2nd derivatives ( Hessian matrix ) is not singular (determinant not equal to zero), the critical points are non-degenerate and geometrically isolated points. The number of negative eigenvalues ​​of the Hessian matrix gives the index. A Morse function is a function with only non-degenerate critical points ("almost all" functions on manifolds are such Morse functions). According to the Morse lemma , the function near the critical point can be represented as a square form, whereby in n dimensions at index r there are r times negative, (nr) times positive signs of the squares.

In the classical Morse theory, conclusions are drawn about the topology of the manifold from the behavior of the Morse function and its critical points. The homotopy type only changes when critical points appear, in such a way that a cell (in the sense of algebraic topology) is "added" to the dimension of the index of the critical point.

Morse inequalities exist between the alternating sum of the numbers of critical points with index and the alternating sum of the rank of homology groups in corresponding dimensions. As a special case, the Euler-Poincare characteristic results as an alternating sum of the numbers of critical points.

Morse had already noticed that the degenerate points are important for transitions in dynamic systems, which was expanded in René Thom's " Catastrophe Theory " .

In 1929 Morse was elected to the American Academy of Arts and Sciences , in 1932 to the National Academy of Sciences, and in 1936 to the American Philosophical Society . In 1933 he received the Bôcher Memorial Prize for his work in analysis. In 1950 he gave a plenary lecture at the International Congress of Mathematicians in Cambridge (Massachusetts) (Recent Advances in Variational Theory in the Large) and also in Zurich in 1932 (The calculus of variations in the large).

Since 1922 he was married to Celeste Phelps, with whom he had a son and a daughter. The marriage ended in 1930 and there was a minor scandal at Harvard in 1932 when his ex-wife married fellow math professor William Fogg Osgood , who was 28 years older than Morse at the time and then had to retire.

He always wanted to be addressed by the first name Marston, his mother's maiden name.

See also

Works

  • Collected Papers , 6 volumes, World Scientific 1987.
  • Raoul Bott (Editor): Selected Papers , Springer 1981.
  • The foundations of the theory of calculus of variations in the large in m-space , Trans. American Mathematical Society 1929.
  • Calculus of variations in the large , American Mathematical Society 1934.
  • Topological methods in the theory of functions of a complex variable , Princeton University Press 1947.
  • Lectures on analysis in the large , 1947.
  • with Stewart Cairns: Critical point theory in global analysis and differential topology , Academic Press 1969.
  • Variational analysis: critical extremals and Sturmian extensions , Wiley 1973, Dover 2007.

literature

  • H. Seifert / W. Threlfall ; Calculus of Variations on a Large Scale [Marston Morse Theory] . [Hamburger Mathematische Einzelschriften, 24th issue]. Leipzig, Teubner, 1938.
  • Raoul Bott: Marston Morse and his mathematical work . In: Bulletin American Mathematical Society, Vol. 3 (1980), pp. 907-950.
  • Stephen Smale : Marston Morse (1892-1977) . In: Mathematical Intelligencer, NF, Vol. 1 (1978), pp. 33f. (Obituary).
  • Joanne E. Snow, Colleen M. Hoover: Mathematician as artist: Marston Morse , Mathematical Intelligencer, Vol. 32 (2010).

Footnotes

  1. ↑ Ulrich Raulff : An American Renaissance: Princeton after the war . In: Frankfurter Allgemeine Zeitung of January 14, 2015, p. N3.
  2. ^ John Willard Milnor : Morse theory . Princeton University Press, Princeton, 5th ed. 1973 (= Annals of mathematics studies , Vol. 51). ISBN 0-691-08008-9 . There the application to the proof of Raoul Bott's periodicity theorem in the theory of the homotopy groups of spheres is given. As a student of Birkhoff, Bott himself was also interested in applications in dynamic systems, and he also tried to apply topology in quantum mechanics.
  3. ^ Member History: Marston Morse. American Philosophical Society, accessed October 31, 2018 .

Web links

personal data
SURNAME Morse, Harold Calvin Marston
ALTERNATIVE NAMES Morse, Marston
BRIEF DESCRIPTION American mathematician
DATE OF BIRTH March 24, 1892
PLACE OF BIRTH Waterville , Maine
DATE OF DEATH June 22, 1977
Place of death Princeton